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DYNAMIC ECONOMETRIC MODELS 
Vol. 9 – Nicolaus Copernicus University – Toruń – 2009 

Elżbieta Szulc 
Nicolaus Copernicus University in Toruń 

Modeling of Dynamic Spatial Processes 

A b s t r a c t. The paper is concerned with econometric modeling of the dynamic spatial 
processes on the example of the GDP per capita in selected European countries. The considera-
tions of the paper are focused on investigations of the structure of components of the spatio-
temporal process. As a result of the analysis some specifications of the dynamic spatial models 
have been obtained. Next the issues of the estimation and verification of the models are presented. 
The main conclusion from the analysis is that the econometric models of the spatio-temporal 
processes ought to be of the dynamic character, e.g. considering the spatial and spatio-temporal 
trends and spatial, temporal and spatio-temporal autodependence as well. 

K e y w o r d s: spatio-temporal trend, autocorrelation, spatial lag model, dynamic spatial model. 

1. Introduction 
 The paper presents the methodology of econometric modeling of the inter-
nal structure of dynamic spatial processes. The considerations recapitulate the 
results of the previous analyses (see: Szulc, 2008, 2009a, 2009b). 
 An empirical illustration of the considerations is the spatio-temporal distri-
bution of the GDP per capita in selected European countries. They are: Austria, 
Germany, the Czech Republic, Slovakia and Hungary. The data relate to the 
established regions according to the European classification system NUTS and 
they are taken from the data released by EUROSTAT. 
 In Szulc (2008) the GDP per capita across the separated area in 2004 was 
analysed. The componential structure of the single “pure” spatial process ( )iZ s , 
observed on the plane at the spatial locations [ ]iii yx ,=s , where i = 1, 2, ..., 84, 
was investigated. In Szulc (2009a) some time aspect was enclosed into the pre-
vious analysis, i.e. the changes of the GDP per capita across the separated area 
in the period: 2000–2006 were considered. The componential structure of the 
spatial process, in successive years, i.e. ( ) [ ]iiiit yxZ ,, =ss , i =1, 2, ..., 84, t =1, 
2, ..., 7, was investigated. Thus the conditional, in relation to time, approach to 



Elżbieta Szulc 

 
18

the analysis of the spatio-temporal process was undertaken. Then the conclu-
sions concerning the total spatio-temporal structure, leading to the appropriate 
empirical model, were formulated only in reference to the so-called spatio-
temporal trend. Moreover some probable specification of the dynamic spatial 
model was proposed. In Szulc (2009b) the approaches, mentioned above, were 
connected with one another by presenting more extended models which de-
scribed the componential structure of the spatio-temporal process ( )tZ i ,s , were 

[ ]iii yx ,=s , i = 1, 2, ..., 84, t = 1, 2, ..., 7. 

 In the investigations the following assumptions were received: 
1. The economic spatial processes demonstrate spatial and/or spatio-temporal 

trends, which are identified as the mean value of the process, changing in 
space and/or in time. 

2. They usually demonstrate autodependence too, which in the structure of the 
process creates the autoregressive component. 

3. The autoregressive component creates the homogeneous/stationary spatial 
or/and spatio-temporal process. 

It means, that for the spatial process ( )iZ s there is assumed the basic structure 
of components, which symbolically may be written down in the following gen-
eral form: 

( ) ( ) ( ) ( ) ( )iiii ZAPZ ssWss ε++= . (1) 

In turn, in the case of the spatio-temporal process ( )tZ i ,s , the basic structure of 
components may be symbolically presented in the form as follows: 

( ) ( ) ( ) ( ) ( )ttZuAtPtZ iiii ,,,,, ssWss ε++= .  (2) 

The symbols in the formulas (1)–(2) signify: 
( )iP s , ( )tP i ,s  – respectively, spatial and spatio-temporal trend which is usually 

expressed in the form of the two-dimensional (three-dimensional) polynomial 
function of the co-ordinates of the location on plane (and of the time variable); 
( )WA , ( )uA ,W  – summable spatial and spatio-temporal shift operators, de-

fined in such a way, that W (the matrix of spatial connections) causes the varia-
ble to be shifted in space, whereas u (the backwards shift operator) causes the 
lag of it in time; ( )isε , ( )ti ,sε  – spatial and spatio-temporal white-noise 
processes. 

2. Investigating the Trend Structure 
 In the investigations of spatial trends the hypothesis of two-dimensional 
polynomial trend was used. The expression of the form: 



Modeling of Dynamic Spatial Processes 

 
19

( ) ∑∑
= =

=
p

k

p

m

m
i

k
imki yxP

0 0
,θs , (3) 

where: [ ]iii yx ,=s  – the co-ordinates of the location on the plane, i = 1, 2, ...,  
N – indexes of the investigated spatial units, k + m ≤ p, presents the spatial trend 
of degree p. 
 The models with the trend of the 1st , 2nd and 3rd degree were estimated and 
verified successively. In all cases the models of the 1st degree appeared the best. 
The results of the estimation and verification of the models are presented in 
Table 1. Finally it was confirmed that the spatial trends occurred in all years of 
the investigated period. 

Table 1. Estimates of the parameters of the spatial trends of the 1st degree for the GDP 
per capita in the period: 2000–2006 

 
Parameters 

Years 
2000 20001 2002 2003 2004 2005 2006 

00θ̂  
15510.4 15943.0 16455.2 17148.2 17925.6 18752.7 19795.4 

( )00θ̂S  
561.122 586.790 597.896 615.903 632.881 680.731 702.129 

00t  27.6418 27.1700 27.5218 27.8424 28.3238 27.5479 28.1934 

10θ̂  
-0.0170 -0.0169 -0.0168 -0.0177 -0.0177 -0.0182 -0.0192 

( )10θ̂S  
0.0017 0.0018 0.0018 0.0019 0.0019 0.0021 0.0021 

10t  -10.000 -9.3890 -9.3333 -9.3158 -9.3158 -8.6667 -9.1429 

01θ̂  
-0.0080 -0.0084 -0.0089 -0.0095 -0.0100 -0.0103 -0.0101 

( )01θ̂S  
0.0024 0.0025 0.0026 0.0027 0.0027 0.0029 0.0030 

01t  -3.3333 -3.3600 -3.4231 -3.5185 -3.7037 -3.5517 -3.3667 
R2 0.5676 0.5450 0.5360 0.5470 0.5355 0.5144 0.5216 

 Figure 1 presents the theoretical surfaces of the trend. Almost parallel loca-
tion of the surfaces show that the spatial trends of the GDP per capita across the 
investigated area in substance do not change with regard to the forms in the 
successive years. The surfaces referring to the consecutive periods are located 
higher and higher in relation to the axis of the GDP per capita values, which 
means, that the mean value of the GDP per capita in the regions grows in time. 
 The results of investigating the spatial trends include the information on the 
spatio-temporal trend of the GDP. Usually such a trend may be described with 
the three-dimensional polynomial function of the following general form: 

( ) ∑∑∑
= = =

=
p

k

p

m

p

l

lm
i

k
ilmki tyxtP

0 0 0
,,, θs , (4) 

where: t – time variable, k + m + l ≤ p, other significations – like in (3). 



 
20

In particu
(P

The mode
successiv
estimates

(19800 145ˆ =θ

Conclusio
The GDP
formed ac
 The e
presented

DG

R2 = 0.55

Figure 1. T
s

3. Inves
 In the
residuals 
pressed b
some, no
observed 

ular, the spati
)ti 000, θ +=s

els of the 1st

ve years may 
 of the con

) ( 412.44616.8
3.7122.512 +

on 1 
P per capita
ccording to t
empirical mo
d by the equa

(
PD ti 88.509, 14367ˆ =

04. 

The surfaces o
spatial trend m

stigating th
e investigati
from the pre

by the formul
t all too high
(see, Table 2

io-temporal t
xi 010100 θθ ++

t  degree tren
be treated as
stants in the

)
t

2
14 . Other p

a across the 
the spatio-tem
odel of the s
ation (6), i.e.:

) ( )
x

0007.02
0177.02.7 −

of the theoret
models in 200

e Autoregr
ons of the s
eviously fitte
la (7), was u
h but statisti
2).  

Elżbieta Szulc

trend of the 
tyi 0010 θ+ .

nd, obtained 
s the conditio
e models sh
parameters al

separated a
mporal trend
patio-tempor
: 

( )
yx ii 0010.0

0093.0−

tical values of
0–2006  

ressive Str
spatial autoc
ed models o

used. In all th
ically signifi

c 

1st degree tak

for the GDP
onal trends i

how the line
lmost do not

area in the i
d of the 1st de
ral trend of 

( )
t

242.113
582.748+

f the GDP per

ructure 
correlation of
of the trend t
he years of t
icant, positiv

kes the form

P spatial proc
n relation to 

ear trend of 
t change. 

investigated 
egree. 
the GDP pe

,

r capita accor

f the 1st ord
the test Mora
the investigat
ve autocorrel

: 
(5) 

cess in the 
time. The 
the form: 

period is 

r capita is 

(6) 

 
rding to the 

der for the 
an’s I, ex-
ted period 
lation was 



Modeling of Dynamic Spatial Processes 

 
21

( )[ ] ( )[ ]

( )[ ]∑

∑∑

∑∑
=

= =

= =

−

−−
= N

i
i

ji

N

i

N

j
ij

N

i

N

j
ij ZZ

ZZZZw

w

NI

1

2

1 1

1 1
s

ss
, (7) 

where: ( )iZ s , ( )jZ s  – values of the process of interest at locations i and j, Z − 
the mean value of the process, wij – the spatial weight of the link between i 
and j. 

Table 2. Testing of spatial autocorrelation 
Year I E(I) Var(I) 
2000 0.234031 -0.012048 0.004699 

Standardized statistic I 
3.6204 

 
p-value = 0.00015 

2001 0.195583 -0.012048 0.004610 
Standardized statistic I 

3.0579 
 

p-value = 0.001114 
2002 0.173191 -0.012048 0.004624 

Standardized statistic I 
2.724 

 
p-value = 0.003225 

2003 0.171139 -0.012048 0.004638 
Standardized statistic I 

2.6899 
 

p-value = 0.003573 
2004 0.164096 -0.012048 0.004647 

Standardized statistic I 
2.584 

 
p-value = 0.004884 

2005 0.138682 -0.012048 0.004637 
Standardized statistic I 

2.2135 
 

p-value = 0.01343 
2006 0.130453 -0.012048 0.004659 

Standardized statistic I 
2.0878 

 
p-value = 0.01841 

Conclusion 2 
The values of the GDP per capita in the neighbouring regions are similar to one 
another. 
In the successive years the values of the Moran’s statistic were decreasing. 
Conclusion 3 
The resemblance among the values of the GDP per capita in the neighbouring 
regions decreases in time. 
 For investigating the spatial range of the autocorrelation two methods were 
used. The first one consisted in calculating and verifying significance of the 
appropriate Moran’s statistics, assuming the neighbourhood of different orders, 
while the second one consisted in using the classic correlation coefficient, cal-
culated for each of the established spatial shift. The significance of the coeffi-



Elżbieta Szulc 

 
22

cients of the 1st and 5th order or of the 1st , 3rd , 4th and even 5th order was con-
firmed (according to the used method)1. 
Conclusion 4 
The spatial autocorrelation of the GDP per capita across the investigated area 
may relate not only to the so-called nearest neighbours. 
 With regard to the diversity of meaning of the results concerning the spatial 
autocorrelation of the higher orders which were obtained with the help of dif-
ferent methods, the autocorrelation of the 1st order was admitted as the most 
possible. 

4. Modeling of the Trend-Autoregressive Structure of the Spatial 
Process 

 The analysis of the trend and autoregressive structure of the GDP per capita 
across the separated area in the successive years led to the following conclu-
sion: 
Conclusion 5 
The following form of the spatial econometric model of the GDP per capita 
should be proposed: 

( ) iiiii GDPyxGDP ερθθθ ++++= W011000 ,  (8) 

the same one for each year of the investigated period. 
 The models of the form (8) are named spatial lag models with regard to the 
presence of the spatial shifted dependent variable W(GDPi). The variable meas-
ures the levels of the investigated phenomenon (of the dependent variable) in 
the neighbouring regions. The results of the estimation and verification of the 
models with the form (8) for the successive years of the period: 2000–2006 are 
presented in Table 3. 
 The obtained empirical models are characterized by significant parameters. 
The residuals of the models do not show any autocorrelation. Thus, it should be 
admitted, that the dependence of the 1st order is sufficient to be taken into ac-
count in the autoregressive structure. 

5. Modeling of the Trend-Autoregressive Structure of the Spatio-
Temporal Process 

 The investigations allow to specify the model referring to the total spatio-
temporal structure of the analyzed process. The successive versions of the spa-
tio-temporal models of the GDP process are presented below. The model of the 

                                                 
1 With regard to the limited volume of the paper (caused by the editorial requirements) the re-

sults of estimations of the appropriate coefficients and of verification of their significance are not 
placed here (for details, see: Szulc, 2009).   



Modeling of Dynamic Spatial Processes 

 
23

form (9) is a direct result of the previous settlements, while the next models 
came into existence by respecification of this model. 

Table 3. The results of the model (8) estimation and verification for the successive years 
of the period: 2000–2006 

Years Parameters Estimates of 
parameters 

Standard errors Statistics z Pr ( > | z | ) 

2000 θ00 7749 1971.2 3.9316 0.000084 
θ10 -0.008797 0.002455 -3.5832 0.000339 
θ01 -0.003669 0.002272 -1.6150 0.106319 

ρ = 0.51452,   test LR = 12.155,    p-value = 0.00049 
Autocorrelation of  residuals: test LM = 0.48187,   p-value = 0.48758 

2001 θ00 8931 2149.3 4.1552 0.00003 
θ10 -0.009725 0.002611 -3.7250 0.000195 
θ01 -0.004394 0.002462 -1.7843 0.074368 

ρ = 0.45339,   test LR = 8.8238,   p-value = 0.0029732 
Autocorrelation of  residuals: test LM = 0.049286,   p-value = 0.82431 

2002 θ00 9734.9 2275.5 4.2781 0.000019 
θ10 -0.010189 0.002680 -3.8015 0.000144 
θ01 -0.004963 0.002565 -1.9348 0.0530169 

ρ = 0.42117,   test LR = 7.2391,   p-value = 0.007133 
Autocorrelation of  residuals: test LM = 0.039654,   p-value = 0.84216 

2003 θ00 10148 2368.9 4.2840 0.000018 
θ10 -0.010728 0.002798 -3.8349 0.000126 
θ01 -0.005286 0.002655 -1.9907 0.046519 

ρ = 0.42057,   test LR = 7.1947,   p-value = 0.007312 
Autocorrelation of  residuals: test LM = 0.021646,   p-value = 0.88303 

2004 θ00 10898 2505.1 4.3505 0.000014 
θ10 -0.010998 0.002855 -3.8526 0.000117 
θ01 -0.005787 0.002763 -2.0943 0.036232 

ρ = 0.40379,   test LR = 6.5691,   p-value = 0.01037 
Autocorrelation of  residuals: test LM = 0.0052268,    p-value = 0.94237 

2005 θ00 12269 2716.7 4.5162 0.000006 
θ10 -0.01215 0.003076 -3.9493 0.000078 
θ01 -0.006447 0.003011 -2.1412 0.032260 

ρ = 0.35696,   test LR = 4.9313,   p-value = 0.026374 
Autocorrelation of  residuals: test LM = 0.15626,   p-value = 0.69262 

2006 θ00 13268 2891 4.5894 0.000004 
θ10 -0.013137 0.003248 -4.0448 0.000052 
θ01 0.006423 0.003096 -2.0746 0.038030 

ρ = 0.33907,   test LR = 4.4004,   p-value = 0.035932 
Autocorrelation of  residuals: LM = 0.12389,   p-value = 0.72486 

5.1. Model with Spatio-Temporal Trend and Spatial Autocorrelation 
 The separated spatial analyses for each point in time and the comparison of 
the obtained results induced to formulate the general conclusion relating to the 



Elżbieta Szulc 

 
24

total spatio-temporal structure of the investigated process in the form of the 
theoretical model as follows: 

( ) titiiiti GDPtyxGDP ,,001010100000, ερθθθθ +++++= W .  (9) 

The results of the estimation and verification of the model (9) are presented in 
Table 4. 

Table 4. The results of the model (9) estimation and verification 
Parameters Estimates of  

parameters 
Standard errors Statistics z Pr (>|z|) 

θ000 
θ100 
θ010 
θ001 

8584.5 
-0.010805 
-0.005248 

447.98 

875.05 
0.001068 
0.001021 
112.72 

9.8103 
-10.1139 
-5.1403 
3.9742 

0.000000 
0.000000 
0.000000 
0.000071 

ρ = 0.41449 
test LR: 49.408, p-value: 0.000000 

Wald statistic: 70.115, p-value: 0.000000 
AIC: 11754 (AIC for lm: 11801) 

Autocorrelation of residuals 
Test LM: 0.003232, p-value: 0.95467 

 The model with the spatio-temporal trend and spatial shifts is characterized 
by significant parameters; the residuals do not show any autocorrelation and it 
is better than the model which takes into consideration only the trend. 

5.2. Model with Spatio-Temporal Trend and with Spatial and also with 
Time Autoregression 

 The existence of the very strong time autocorrelation of the GDP per capita 
(the coefficient of time autocorrelation of the 1st order for the residuals of the 
model with the spatio-temporal trend of the 1st degree equals 0.9951) justifies 
including the component GDPi,t-1 into the model which describes the structure 
of the GDP process. Thus, the next specification of the model is following: 

( ) tititiiiti GDPGDPtyxGDP ,,1,001010100000, εραθθθθ ++++++= − W .(10) 

The results of the estimation and verification of the model (10) are presented in 
Table 5. 

Apart from the improvement in the general degree of the model fitting, it cannot 
be treated as the final one because the autocorrelation appeared in the residuals. 

5.3. Model with the Spatio-Temporal Trend and with Spatial, Time and 
Spatio-Temporal Autoregression 

 Just as the coefficients of the spatial and time autocorrelation, the coeffi-
cient of the spatio-temporal autocorrelation of the 1st order appeared significant. 
Its value amounted to 0.1636. Therefore the next model of the GDP spatio-



Modeling of Dynamic Spatial Processes 

 
25

temporal structure additionally takes into consideration the component 
W(GDPi,t-1). It has the following form: 

( ) ( ) .,1,,

1,001010100000,

tititi

tiiiti

GDPGDP
GDPtyxGDP
εγρ

αθθθθ

+++

++++=

−

−

WW
  (11) 

Table 5. The results of the model (10) estimation and verification 
Parameters Estimates of  

parameters 
Standard errors Statistics z Pr (>|z|) 

θ000 
θ100 
θ010 
θ001 
α 

161.68 
-0.000219 
-0.000305 

125.35 
1.0420 

121.88 
0.000145 
0.000121 
13.627 

0.004225 

1.3265 
-1.5097 
-2.5191 
9.1985 

246.6556 

0.18467 
0.13111 
0.01176 
0.00000 
0.00000 

ρ = -0.034921 
test LR: 17.139, p-value: 0.000000 

Wald statistic: 17.519, p-value: 0.000000 
AIC: 7643.4 

Autocorrelation of residuals 
Test LM: 49.851, p-value: 0.000000 

The results of the estimation and verification of the model (11) are presented in 
Table 6. 

Table 6. The results of the model (11) estimation and verification 
Parameters Estimates of  

parameters 
Standard errors Statistics z Pr (>|z|) 

θ000 
θ100 
θ010 
θ001 
α 
γ 

252.99 
-0.000290 
-0.000275 

78.610 
1.0458 

-0.42254 

115.64 
0.000137 
0.00014 
13.9317 
0.004019 
0.057249 

2.1879 
-2.1088 
-2.4146 
5.6485 

260.1819 
-7.3808 

0.02868 
0.03496 
0.01575 
0.00000 
0.00000 
0.00000 

ρ = 0.37102 
test LR: 37.425, p-value: 0.000000 

Wald statistic: 44.667, p-value: 0.000000 
AIC: 7600.3 

Autocorrelation of residuals 
Test LM: 0.12572, p-value: 0.72291 

 The considered model is characterized by significant parameters. The resi-
duals of the model do not show any autocorrelation. According to its general 
fitting to the data it is the best among all the models proposed in this paper. 



Elżbieta Szulc 

 
26

5. Final Remarks 
 The considerations of the paper confirm that investigating the properties and 
structures of spatial and spatio-temporal economic processes is important for 
modeling of them. 
 The econometric models of the spatio-temporal processes should have the 
dynamic character. It is expressed in the appropriate specification of the trend-
autoregressive structure, characterizing temporal, spatial and spatio-temporal 
tendencies and the lags and spatial or/and spatio-temporal shifts of the observed 
dependence. 
 The GDP per capita across the separated area in the investigated period 
realizes the spatio-temporal process, which shows the spatio-temporal trend and 
the spatial and spatio-temporal autodependence. The specification of the dy-
namic spatial model for the GDP per capita caused that a “good” empirical 
model was obtained. 

References  
Kopczewska, K. (2006), Ekonometria i statystyka przestrzenna z wykorzystaniem programu  

R CRAN (Spatial Econometrics and Statistics with R CRAN), CeDeWu Sp. Z o. o. 
Schabenberger, O., Gotway, A. C. (2005), Statistical Methods for Spatial Data Analysis, Cham-

pion & Hall/CRC, New York. 
Szulc, E. (2008), Analiza struktury ekonomicznych procesów przestrzennych na przykładzie PKB 

w wybranych krajach europejskich (Analysis of the Structure of Economic Spatial 
Processes on the Example of GDP in Chosen European Countries), Acta Universitatis Ni-
colai Copernici, Ekonomia (Economy) XXXVIII, No. 388, 7–20.  

Szulc, E. (2009a), Analiza zmian w czasie struktury ekonomicznych procesów przestrzennych na 
przykładzie PKB w wybranych krajach europejskich (Analysis of the Changes in Time of 
the Structure of Economic Spatial Processes on the Example of GDP in Chosen European 
Countries), a report presented during the 3rd Professor Aleksander Zeliaś Scientific Confe-
rence on Modelling and Forecasting of Socio-Economic Phenomena, Zakopane, May 5th – 
8th 2009, in print in: J. Pociecha (ed.), Współczesne problemy modelowania i prognozowa-
nia zjawisk społeczno-gospodarczych (Modern Problems of Modelling and Forecasting of 
Socio-Economic Phenomena), Studia i Prace Uniwersytetu Ekonomicznego w Krakowie 
(Studies and Works of Economic University in Cracow). 

Szulc, E. (2009b), Modelowanie dynamicznych procesów przestrzennych (Modeling of Dynamic 
Spatial Processes), Acta Universitatis Nicolai Copernici, Ekonomia (Economy) XXXIX, 
No. 389, 63–70. 

Modelowanie dynamicznych procesów przestrzennych 
Z a r y s  t r e ś c i.  Artykuł przedstawia ekonometryczną analizę procesu przestrzenno-
czasowego na przykładzie PKB w wybranych krajach europejskich. Przedmiotem rozważań są 
przestrzenne i przestrzenno-czasowe trendy oraz autozależności charakteryzujące składnikową 
strukturę badanego procesu. Składniki te są podstawą do specyfikacji dynamicznych modeli 
przestrzennych. Zaproponowane w artykule specyfikacje dynamicznych modeli przestrzennych 
poddaje się empirycznej weryfikacji. 

S ł o w a  k l u c z o w e: trend przestrzenno-czasowy, autokorelacja, model przesunięć prze-
strzennych, dynamiczny model przestrzenny. 


